On the Existence of Solutions of Nonlinear Boundary Value Problems for a System of Differential Equilibrium Equations for Timoshenko-Type Shells in Isometric Coordinates

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We prove the existence of solutions of a boundary value problem for a system of five nonlinear second-order partial differential equations with given nonlinear boundary conditions, which describes the equilibrium state of elastic shallow inhomogeneous isotropic shells with free edges in the Timoshenko shear model referred to isometric coordinates. The boundary value problem is reduced to a nonlinear operator equation for generalized displacements in the Sobolev space, the solvability of which is established using the contraction mapping principle.